Monk's 'Startling' Math Discovery

An astonishing instance of mathematical anticipation has recently come to light, showing that a monk discovered a complex formula centuries earlier than previously thought.

The Mandelbrot set is one of those dizzying fractal shapes that
have come to symbolize the modern science of chaos theory. Like all such
shapes, it is indefinitely convoluted and has arcs giving rise to smaller arcs and flares branching into smaller flares.

Oriented vertically, it can be viewed as a star, and it is this way of seeing the set that leads to what is probably one of the most arresting discoveries in the history of mathematics.

In his wonderfully multitudinous Web site, The Apothecary’s Drawer (see Web link, left), Roy Girvan, an English
science writer and Web designer, tells the story of a retired mathematics professor traveling in Germany who was shocked to see in a medieval religious manuscript a nativity scene with a Star of Bethlehem in the unmistakable shape of the Mandelbrot set.

Bill Gates in the Dead Sea Scrolls

The professor, Bob Schipke, confessed to being stunned. “It was like finding a picture of Bill Gates in the Dead Sea Scrolls. The title page named the copyist as Udo of Aachen, and I just had to find out more about this guy.”

After investigation, Schipke discovered that this 13th century monk, previously known only for his poetry and essays, had discovered the simple secret for the generation of the Mandelbrot set 700 years before Benoit Mandelbrot!

Happily, some of Udo’s notebooks have survived, and Schipke found
that the genius monk wrote knowledgeably about the notion of probability. He also discovered that the monk had discovered a method for estimating the
value of pi that had previously been thought to have been first employed by 18th century naturalist Comte de Buffon.

But Udo of Aachen’s most staggering feat is the construction of the
Mandelbrot set, which involves the multiplication of complex numbers.

These
are numbers of the form a + bi, where a and b are what we normally think of
as real numbers, and i, a so- called imaginary number, is the square root
of -1.

Spiritual, Profane and Complex

Incredibly these numbers would not become a part of mathematical
practice for another 500 years, but Udo of Aachen wrote of the
intertwined spiritual and profane parts of all entities, including numbers,
and through some scrim of theological reasoning, he concluded that the
product of two purely profane numbers was a negative spiritual number, just
as i² = -1.

In this way Udo derived rules for dealing with profane/spiritual
numbers that were essentially identical to those used today for manipulating
complex numbers.

Using these rules, more theology (a number multiplied by
itself is somehow the analogue of spiritual reflection), and repeatedly
substituting complex numbers into the expression z² + c to determine their
“fate” eventually led Udo of Aachen to the Mandelbrot set.

Prof. Schipke, with the help of historian Antje Eberhardt at
the University of Munich, wrote up his findings on Udo in the March 1999
issue of the Harvard Journal of Historical Mathematics.

Few Comparable Finds

But despite the prestige
of the publication, the discovery has not received the attention it
warrants, and this may be one reason that Ray Girvan has so beautifully told
the story on his web site and is certainly why I am publicizing it here.

Not since the Piltdown-Sokal Theorem was discovered to have first
been proved in 1907 by a provincial Russian electrical engineer, and not in
1979 by Piltdown and Sokal as originally had been thought, has the date of a
first proof been so misjudged.

There is only one case of misattribution in
the history of mathematics that is of comparable magnitude. The famous
triangular array of numbers discovered by Pascal in the 17th century
was anticipated five centuries before him by the Chinese mathematician Chia
Hsien.

Even this example, however, is trumped by the 700 year differential
between Udo’s and Mandelbrot’s discovery.

Professor of mathematics at Temple University, John Allen Paulos is the author of several best-selling books, including Innumeracy and A Mathematician Reads the Newspaper. His Who’s Counting? column on ABCNEWS.com appears on the first day of every month.